@dg: Great question. When dealing with continuous growth, we can mix and match rate and time as we please. Here’s why.
Let’s say we have 1 year of 30% continuous growth. That means each individual dollar intends create interest of 30% when all is said and done. Along the way, its interest will start growing, but the original dollar doesn’t know! The 30% growth (from a continuous case) means “An individual dollar, Mr. Blue, knows he will create a Mr. Green that is 30% as large at the end of the period. What Mr. Green happened to be doing along the way is none of Mr. Blue’s business.”
So, with this model, it doesn’t matter if Mr. Blue builds up Mr. Green in one year of 30%, or in 30 years of 1% progress each year. At the end, Mr. Blue has made a Mr. Green who is 30% as large as he is.
Clearly, it’s better for us, the investors, to get the growth in 1 year vs. 30, but either approach results in the same total amount (e^{30% * 1} = e^{1% * 30} = e^{.30} = 1.35. Either way, have a total growth from 1 to 1.35. The extra .05 is because our interest did work along the way. (We’re assuming the growth happens in whatever time is necessary, then stops afterwards.)
@vivek: Practice problems are key, there’s some at Khan Academy or http://www.exampleproblems.com/.
@chris: Great question. When there’s multiple terms it helps to break them apart:
$$e^{-(ax + bx^2)} = e^{-ax} \cdot e^{-bx^2}$$
Basically, we have two decay factors acting simultaneously (they are decay factors, not growth factors, since the exponent is negative, and will be shrinking our value).
The first is a regular decaying exponential function, e^{-ax}. The second is an exponential function which is shrinking extremely quickly – with the square of time.
The general term e^{-bx^2} is a Gaussian function (bell curve) [http://en.wikipedia.org/wiki/Gaussian_function] and is basically a decaying exponential on steroids. Instead of a nice sloping curve it will be shrinking extremely quickly. What a regular exponential would see in 100 time periods, it sees in 10. I’m not that familiar with this function (or stats in general) but would be a fun thing to investigate later on.